Mathematical Methods of Physics/Linear Algebra

The simplest structures on which we can study operations of both "algebra" and "calculus" is the Banach space. The crucial importance of Hilbert Spaces in Physics is due to the fact that the not only are Hilbert Spaces a special case of Banach space, but also because they contain the idea of inner product and the related conjugate-symmetry. (This chapter requires some familiarity with basic measure theory)

In the examples of Hilbert spaces given below, the underlying field of scalars is the complex numbers C, although similar definitions apply to the case in which the underlying field of scalars is the real numbers R.

for all x and y in ℓ2(B){\displaystyle \ell ^{2}(B)}. B does not have to be a countable set in this definition, although if B is not countable, the resulting Hilbert space is not separable. Every Hilbert space is isomorphic to one of the form ℓ2(B){\displaystyle \ell ^{2}(B)} for a suitable set B. If B=N, the natural numbers, this space is separable and is simply called ℓ2{\displaystyle \ell ^{2}}.

Among examples of Hilbert spaces, the one that holds the most interest for the physicists are the L2{\displaystyle {\mathcal {L}}^{2}} spaces.

Consider L2{\displaystyle {\mathcal {L}}^{2}} to be the set of all functions f:[a,b]→C{\displaystyle f:[a,b]\to \mathbb {C} } that are square integrable with respect to a real measure μ{\displaystyle \mu }, that is ∫ab∥f∥2dμ{\displaystyle \int _{a}^{b}\|f\|^{2}d\mu } is well-defined.

Provided the inner product exists for any pair of functions f,g{\displaystyle f,g}, we can see that L2{\displaystyle {\mathcal {L}}^{2}} is an inner product space.

The reader may notice an ambiguity here, as (f∗g)=(f∗g′){\displaystyle (f*g)=(f*g')} need not imply that g=g′{\displaystyle g=g'}. To resolve this, we use a different equivalence relation between functions, f∼f′⇔∫ab∥f−f′∥dμ=0{\displaystyle f\sim f'\Leftrightarrow \int _{a}^{b}\|f-f'\|d\mu =0}, and hence, f=f′{\displaystyle f=f'} at all points of [a,b]{\displaystyle [a,b]} except for a set of points of measure 0{\displaystyle 0}.

The L2{\displaystyle {\mathcal {L}}^{2}} space is an example of what are called the Lp{\displaystyle {\mathcal {L}}^{p}} spaces. It can be shown([1]) that all Lp{\displaystyle {\mathcal {L}}^{p}} spaces are complete, and hence, the Lebesgue space, L2{\displaystyle {\mathcal {L}}^{2}} is also complete.

Thus, we have that L2{\displaystyle {\mathcal {L}}^{2}} is a Hilbert space.